This invention relates to lasers, and more particularly to passively mode-locked solid state lasers designed to operate at repetition rates exceeding 1 GHz.
Solid-state lasers are known in the art. Their laser gain media are dopant ions incorporated in dilute concentrations in solid hosts. The laser gain medium can be optically excited to emit electromagnetic radiation by impinging a pumping beam on the laser gain medium. High-repetition rate lasers are desirable for a number of applications, such as for use as seed sources for driving radio-frequency photocathodes. These RF photocathodes are then used to inject high-energy electron bunches into a linear accelerator. It is often desirable to have the laser repetition rate operating at the drive frequency of the linear accelerator, which is typically at 2.8 GHz or higher. It is also possible to use high-repetition rate lasers synchronized to the drive frequency of the accelerator in diagnostic tools or in optical-electron interactions after the electrons are fully accelerated.
Other possible applications of high-repetition rate lasers are in the area of telecommunications, photonic switching, and optoelectronic testing. As networks and electronic components continue to increase in terms of bandwidth and clock frequency, optical pulsed laser sources become more important for driving, sensing, and testing of these components. One example of this application for optical clocking of integrated circuits is disclosed in U.S. Pat. No. 5,812,708 (Rao).
Mode locking is a special operation regime of lasers where an intra-cavity modulation (amplitude or phase modulator) forces all of the laser modes to operate at a constant phase, i.e., phase-locked or xe2x80x9cmode-lockedxe2x80x9d, so that the temporal shape of the laser output forms a continuously repeating train of short (typically in the range of picoseconds or femtoseconds) optical pulses. The repetition rate of this pulse train is set by the inverse of the laser round-trip time, or equivalently by the free spectral range of the laser, frep=c/2L where c is the speed of light and L is the cavity length for a standing wave cavity. This repetition rate frep is termed the fundamental repetition rate of the laser cavity, since this corresponds to only one laser pulse circulating in the cavity per round trip. The repetition rate can be scaled by integer multiples N of the fundamental repetition rate under certain conditions, and this is called harmonic mode locking. In this case, there are multiple laser pulses circulating in the cavity per round trip. The minimum possible pulse width of the laser is nominally set by the line width of the laser transition, following approximately the condition that tminxe2x89xa70.44/xcex94f where xcex94f is the line width of the laser transition. For typical laser materials such as Nd:YAG or Nd:vanadate, the laser line width can support pulses to less than 10 ps. For broader-bandwidth materials such as Nd:glass or Ti:sapphire, pulse widths to below 100 fs and even below 10 fs can be generated.
Mode locked lasers are well known in the state-of-the-art, having been first described in the 1960""s (see H. W. Mocker et al., xe2x80x9cMode competition and self-locking effects in a Q-switched ruby laser,xe2x80x9d Applied Physics Letters, vol. 7, pp. 270-273, 1965). Passive mode locking using a saturable absorber was discovered almost immediately thereafter. Most mode-locked lasers have used active modulators, where the term xe2x80x9cactivexe2x80x9d means that a source of power such as a radio-frequency signal or another electronic signal must be periodically applied to the modulator. Typical active modulators are acousto-optical modulators (AOMs, Bragg cells) or electro-optical (EOMs, Pockels cells). Active modulators can modulate the amplitude (AOMs or EOMs) or the phase (EOMs) of the optical signal to achieve mode locking.
Active mode lockers have the disadvantages of cost and complexity. A typical device requires a precision electro-optical component, plus drive electronics which typically consists of high-power, high-stability RF-signal (for AOMs) or high-voltage (for EOMs) components. Additionally, feedback electronics may be required to stabilize either the drive signal for the modulator and/or the laser cavity length to achieve the necessary stability from the system (cf. U.S. Pat. No. 4,025,875, Fletcher et al., xe2x80x9cLength controlled stabilized mode-lock Nd:YAG laserxe2x80x9d, and Lightwave Electronics, Series 131 data sheet, March 1994).
Active mode locking has been available in commercial lamp-pumped laser systems and more recently in diode-pumped laser systems at repetition rates typically of 100 MHz and extending up to 250 MHz. Research on active mode locking has been done on higher repetition rates, achieving repetition rates of approximately 2 GHz (see K. J. Weingarten et al., xe2x80x9cTwo gigahertz repetition rate, diode-pumped, mode-locked Nd:YLF laserxe2x80x9d, Optics Letters, vol. 15, pp. 962-964, 1990), 5 GHz (P. A. Schulz et al., xe2x80x9c5-GHz mode locking of a Nd:YLF laserxe2x80x9d, Optics Letters, vol. 16, pp. 1502-1504, 1991), 20 GHz (A. A. Godil et al., xe2x80x9cHarmonic mode locking of a Nd:BEL laser using a 20-GHz dielectric resonator/optical modulatorxe2x80x9d, Optics Letters, vol. 16, pp. 1765-1767, 1991), and more recently 40 GHz (A. J. C. Viera et. al., xe2x80x9cMicrochip laser for microwave and millimeter-wave generationxe2x80x9d, IEEE MTT-S IMOC""97 Proceedings). In all cases the systems required an active modulator driven by a stable RF source and an RF amplifier. The highest repetition rates at 40 GHz were achieved with xe2x80x9charmonicxe2x80x9d mode locking (see M. F. Becker et al., xe2x80x9cHarmonic mode locking of the Nd:YAG laserxe2x80x9d, IEEE Journal of Quantum Electronics, vol. QE-8, pp. 687-693, 1972), where the modulator is driven at some integer multiple of the fundamental laser repetition rate. This is an additional source of complexity and instability in the laser system. In general we wish to avoid harmonic mode locking if possible.
It is also possible to generate high repetition rates using other laser medium such as rare-earth-doped fiber lasers, and semiconductor lasers. Repetition rates of greater than 10 GHz have been demonstrated in semiconductor quantum well lasers (see U.S. Pat. No. 5,040,183, Chen et al., xe2x80x9cApparatus comprising optical pulse-generating meansxe2x80x9d), achieving pulse repetition rates even to greater than 100 GHz. However, their approach appears to be limited in terms of average power. Fiber lasers have also been demonstrated to high repetition rates using active or harmonic passive mode locking (see U.S. Pat. No. 5,414,725 Fermann et al., xe2x80x9cHarmonic partitioning of a passively mode-locked laserxe2x80x9d, and S. V. Chernikov et al., xe2x80x9cDuration-tunable 0.2-20 ps 10-GHz source of transform-limited optical pulse based on an electro-absorption modulatorxe2x80x9d, Optics Letters, vol. 20, pp. 2399-2401, 1995) Passive mode locking at the fundamental repetition rate, on the other hand, is a much simpler, robust, and lower-cost approach to generating mode-locked pulses. Passive mode locking is also well-established in the state of the art (see A. J. DeMaria et al., xe2x80x9cSelf mode-locking of lasers with saturable absorbersxe2x80x9d, Applied Physics Letters, vol. 8, pp, 174-176, 1966). The most significant developments in passive mode locking in the recent years have been Kerr-Lens Mode locking (KLM) (U.S. Pat. No. 5,163,059, Negus et al., xe2x80x9cMode-locked laser using non-linear self-focusing elementxe2x80x9d) for generation of femtosecond pulses from Ti:sapphire and other femtosecond laser systems, and the semiconductor saturable absorber mirror (SESAM) device for generating picosecond and femtosecond pulses in a wide number of solid-state lasers (see U. Keller et al., xe2x80x9cSemiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,xe2x80x9d Journal of Selected Topics in Quantum Electronics (JSTQE), vol. 2, no. 3, pp. 435-453, 1996). Passive mode locking relies on a saturable absorber mechanism, which produces either decreasing loss with increasing optical intensity, or similarly an increase gain with increasing optical intensity. When the saturable absorber parameters are correctly adjusted for the laser system, the optical intensity in the laser cavity is enhanced such that a mode-locked pulse train builds up over a time-period corresponding to a given number of round-trips in the laser cavity. Most passively mode locked lasers have been operated at repetition rates of approximately 100 MHz, corresponding to a cavity length of approximately 1.5 m. This cavity length is appropriate for many applications (such as seeding a regenerative laser amplifier) and is also convenient for building laboratory-scale lasers. Work has been done to achieve higher repetition rates, which could be important for telecommunications and optical clocking applications (see U.S. Pat No. 4,930,131, Sizer, xe2x80x9cSource of high repetition rate, high power optical pulsesxe2x80x9d, U.S. Pat. No. 5,274,659, Harvey, et. al., xe2x80x9cHarmonically mode-locked laserxe2x80x9d, U.S. Pat. No. 5,007,059, Keller et al., xe2x80x9cNonlinear external cavity mode locked laserxe2x80x9d; B. E. Bouma et al., xe2x80x9cCompact Kerr-lens mode-locked resonatorsxe2x80x9d, Optics Letters, vol. 21, 1996, pp. 134-136; and B. C. Collings et al, xe2x80x9cTrue fundamental solitons in a passively mode-locked short-cavity Cr4+:YAG laserxe2x80x9d, Optics Letters, vol. 22, pp. 1098-2000, 1997). However, passive mode locking in solid-state lasers has not been readily achieved at fundamental repetition rates beyond 1 GHz. There are a number of reasons for this limitation. First, for a given average power, the pulse energy and thus the peak power in a pulse will decrease as the laser repetition rate increases (given that the pulsewidth also stays constant). For laser relying on peak-power induced non-linearities to achieve passive mode locking (i.e., lasers using KLM) it becomes increasingly difficult to mode-lock at higher repetition rates. In addition, the cavity size decreases in length inversely proportional to the repetition rate, and it becomes more difficult to adequately provide dispersion compensation. As noted, solid-state lasers using KLM have not been reported substantially beyond repetition rates of 1 GHz (see B. E. Bouma et al., xe2x80x9cCompact Kerr-lens mode-locked resonatorsxe2x80x9d, Optics Letters, vol. 21, 1996, pp.134-136, and U.S. Pat. No. 5,553,093 Ramaswamy et. al., xe2x80x9cDispersion-compensated laser using prismatic end elementsxe2x80x9d).
For passively mode locked lasers using SESAMs for mode locking, the limitation on repetition rate is the onset of Q-switching instabilities (see U. Keller et al., xe2x80x9cSemiconductor saturable absorber mirrors (SESAMs) for femtosecond to nanosecond pulse generation in solid-state lasers,xe2x80x9d Journal of Selected Topics in Quantum Electronics (JSTQE), vol. 2, no. 3, pp.435-453, 1996, and U. Keller, xe2x80x9cUltrafast all-solid-state laser technologyxe2x80x9d, Applied Physics. B, vol. 58, pp.347-363, 1994). This has also limited the laser repetition rate to the range of several hundred megahertz typically. Using the technique of coupled cavity mode locking (RPM), a repetition rate of 1 GHz was demonstrated (see U. Keller, xe2x80x9cDiode-pumped, high repetition rate, resonant passive mode-locked Nd:YLF laserxe2x80x9d, Proceedings on Advanced Solid-State Lasers, vol. 13, pp.94-97, 1992). However this is a much more complicated laser due to the additional laser cavity which has to be carefully aligned with the main laser cavity.
It would be advantageous to achieve repetition rates greater than 1 GHz for many applications such as synchronization with linear particle accelerators (which typically operate at 3 GHz or higher), use in high-speed telecommunications networks as optical pulse sources, and optical clocking of circuits and system in the gigahertz range. These lasers may also find applications in measurement applications such as precision ranging, optical testing of photodetectors and other optically triggered components, and electro-optical test methods on electronics and integrated circuits.
An object of this invention to provide a simple, robust, passively mode locked laser with repetition rates of greater than 1 GHz, extending as high as possible (up to approximately 50 GHz or higher). It is a further object of the invention to provide a laser system which generates a relatively large average power of 100 mW and higher, which is useful for a number of optical probing and detection applications, in a beam which is substantially a fundamental spatial mode, and which is pumped by a semiconductor laser (xe2x80x9cdiode-pumpedxe2x80x9d) so that it is compact, efficient, and low-cost.
Yet another object is to accomplish the foregoing with semiconductor saturable absorber mirrors (SESAMs) which further simplifies the design.
According to the invention, a passively mode-locked solid-state laser can be caused to emit a continuous-wave train of electromagnetic-radiation pulses characterized by an effective wavelength, the fundamental repetition rate of the emitted pulses exceeding 1 GHz, without Q-switching, under certain conditions described below. The laser comprises an optical resonator, a solid-state laser gain element placed inside said optical resonator, means for exciting said laser gain element to emit electromagnetic radiation characterized by the effective wavelength, and means for passive mode locking comprising a saturable absorber. The laser gain element preferably consists of a laser material with a stimulated emission cross section exceeding 0.8xc3x9710xe2x88x9218 cmxe2x88x922 at the effective wavelength; typically, the laser gain element is made of Nd:vanadate. The saturable absorber is preferably a semiconductor saturable absorber mirror device.
The method for emitting a continuous-wave train of electromagnetic-radiation pulses characterized by an effective wavelength, the pulses being emitted with a fundamental repetition rate exceeding 1 GHz, comprises the steps of exciting a laser gain element to emit electromagnetic radiation characterized by the effective wavelength, said laser gain element being placed inside an optical resonator, recirculating said electromagnetic radiation in said optical resonator, and passively mode locking said electromagnetic radiation using a saturable absorber. When the conditions necessary to avoid the Q-switching instabilities in passively mode-locked lasers are examined more carefully, the following stability condition can be derived:
(Flaser/Fsat,laser)xc2x7(Fabs/Fsat,abs) greater than xcex94Rxe2x80x83xe2x80x83(1)
where Flaser is the fluence in the laser material, Fsat,laser=h"ugr"/"sgr"laser is the saturation fluence of the laser material, h is Planck""s constant, "ugr" is the center laser frequency, "sgr"laser is the laser cross-section parameter (see W. Koechner, Solid-State Laser Engineering, 4th Edition, Springer-Verlag New York, 1996), Fabs is the fluence on the absorber device, Fsat,abs=h"ugr"/"sgr"abs-eff is the effective saturation fluence of the absorber, where "sgr"abs-eff is the effective cross-section parameter of the absorber device, and xcex94R is the modulation depth of the absorber device. This equation can be used to scale a laser for operation at higher repetition rates. If all else remains constant (i.e., mode size in laser material and on the absorber, average power, and pulsewidth), as the repetition rate increases, the left-hand term decreases due to decreasing pulse energy. It is possible to avoid Q-switching under this condition by arbitrarily decreasing the modulation depth xcex94R. However, below a certain modulation depth, the absorber will not have a strong enough effect to start and sustain mode locking.
For further clarity we simplify Eq. (1) to the following:
Slaserxc2x7Sabs greater than xcex94Rxe2x80x83xe2x80x83(2)
where Slaser is the fluence ratio in the laser material, and Sabs is the fluence ratio on the absorber. This reduced notation allows us to simplify the further discussion. To achieve the maximum figure of merit, one can change the laser design to increase the fluence ratio Slaser in the laser material, or to increase the fluence ratio Sabs in the absorber.
First, we consider increasing the absorber fluence ratio Sabs. There are two limits to the fluence level on the absorber. First, a very high fluence can result in optical damage. Damage levels of SESAM absorbers have been measured in the range of 30 mJ/cm2. Secondly, a very high fluence (but still below the damage threshold) may cause the laser to operate with multiple pulses per round trip (i.e., a form of harmonic mode locking). This may be desirable as a method to increase the repetition rate of the laser, however, it may result in decreased operation stability of the laser. Typical fluence levels on the SESAM can range from approximately Fsat,abs to as high as 50xc2x7Fsat,abs. (representative saturation fluence Fsat,abs are approximately 50 to 100 xcexcJ/cm2). It is also important to consider the saturation fluence of the SESAM, Fsat,absxe2x88x92eff. This parameter is effectively set by the semiconductor absorber material cross-section value ("sgr"abs). Although it is possible to change the effective saturation fluence of the SESAM device by scaling the design so that the absorber sees a different field intensity, this will not allow us to increase the fluence ratio above the limits set by material damage or multiple pulsing. Possible methods to reduce the absorber saturation fluence would be to use the exciton effect. Note that it is possible to tune the exciton effect, which has a relatively narrow optical frequency range, by temperature tuning the material. It is thus possible to optimize the modulation depth of the SESAM device by temperature tuning the entire SESAM device while it is in the laser cavity to maximize its modulation depth. This is advantageous as it allows us to fine tune the passive mode-locking start-up for maximum repetition rate, i.e., we can use a device with a low modulation depth, but tuned so that it just provides enough modulation to start the passive mode locking, but not enough to start Q-switching the laser.
Another possible method to reduce the saturation fluence is to use a different absorber material with a fundamentally different cross-section value. Currently this is restricted due to the fabrication nature of the SESAM devices, which typically relies on InGaAs or similar semiconductor combinations with various doping levels of the Indium concentration to achieve absorption at the desired wavelength. This material system has a roughly constant saturation cross section.
There are a number of material parameters that can be optimized to improve the SESAM fluence ratio. First, by proper doping of the SESAM absorber, the modulation depth can be increased for a given absorber thickness, conversely allowing for a shorter absorber for maintaining a constant modulation depth, which allows for reduced fixed loss, which results in more efficient laser operation. Secondly it is possible to passivate the surface of the SESAM device to improve its damage threshold, allowing for operation of the SESAM with a higher fluence.
There are also a number of techniques to optimize the design of the SESAM device for maximizing the repetition rate of the laser. Let us review the basic design issues with SESAMs. Basically they are a combination of non-absorber dielectric layers which are typically arranged in quarter-wave and half-wave layers to form a mirror structure. The absorber layer can be imbedded in any of the quarter-wave or half-wave layer structures, as long as the entire filter structure is properly designed, without substantially degrading the reflectivity of the mirror structure. Note that it is also possible to design the mirror structure with xe2x80x9cchirpedxe2x80x9d layer thicknesses to introduce increased operating bandwidth or to introduce some dispersive function of the mirror.
The position of the absorber in the structure can play a key role in setting the device parameters. Basically the saturation fluence of the device is set by the formula
Fsat,eff=Fsat,mat/xcex6xe2x80x83xe2x80x83(3)
where Fsat,eff is the effective saturation fluence of the SESAM, Fsat,mat is the saturation fluence of the absorber material, and xcex6 is a finesse factor (see U. Keller, xe2x80x9cUltrafast all-solid-state laser technologyxe2x80x9d, Applied Physics. B, vol. 58, pp. 347-363, 1994) given by the SESAM design. There are several ways to influence Fsat,eff. By positioning the absorber in a region of low optical field, we can increase the effective saturation fluence of the device, and correspondingly reduce the modulation depth. The modulation depth can be independently adjusted by the absorber thickness. Note that as the absorber thickness becomes comparable to the length of the standing optical wave in the device, the effective saturation fluence will also start to increase. One technique to increase the modulation depth while maintaining a lower effective saturation fluence is to position multiple absorber layers at the peak of the standing wave in more than one of the appropriate layers.
Note it is also possible to increase the effective saturation fluence of the SESAM by adding reflective layers on top of the layer holding absorber. Since certain dielectrics have higher damage threshold than semiconductor materials, it may be advantageous to have the top layers of the structure with damage-resistant dielectrics, then the absorber layer, then a semiconductor dielectric structure underneath.
It is also possible to improve the damage threshold of semiconductor devices by appropriate passivation of the top layer. This passivation layer prevents oxygen and other contaminants from migrating into the semiconductor structure, and also holds in place any contaminants that may already exist on the face of the device. At the same time, the passivation can be made very thin so that it is optically transparent and does not substantially affect the reflectivity and absorption structure of the device. A typical passivation layer for example would be to deposit 2 nm of Si on the final face of the SESAM device before it has been removed from its fabrication chamber and exposed to possible contaminants. Passivation techniques for semiconductor laser devices have been disclosed by U.S. Pat No. 5,144,634, Gasser et. al., xe2x80x9cMethod for mirror passivation of semiconductor laser diodexe2x80x9d.
Next we consider increasing the fluence ratio in the laser material. The main limit to increasing the fluence of the laser beam in the crystal will be limited by mode-matching requirements set by the pump laser (see D. Kopf et al., xe2x80x9cHigh-average-power diode-pumped femtosecond Cr:LiSAF lasersxe2x80x9d, Applied Physics B, vol. 65, pp. 235-243, 1997). Although it is possible that damage to the laser crystal could occur at very high fluence levels, normally we do not operate near to this limit (approximately 100 mJ/cm2 for a 10 ps pulse in Nd:YAG). However in contrast to the saturation fluence in the SESAM, we can change the saturation fluence by changing the laser crystal. Typical laser crystals used in the past have included Nd:YAG, Nd:YLF, and Nd:vanadate. Table I shows representative values of emission cross-sections of various neodymium hosts at approximately 1064 nm. Comparing the laser cross-section a as given in Table I, we see that Nd:vanadate has a substantially higher cross-section, and therefore a lower saturation fluence. Therefore this crystal is one of the best choices for minimizing the Q-switching figure of merit (FOM) compared to other typical Nd-doped crystals.
The next consideration is the fluence of the laser mode in the laser crystal. Here the key limitation is given by the conditions set by proper mode-matching. For efficient optical pumping, the overlap of the pumping beam with the laser mode must generally be high over the absorption length in the crystal. For this purpose, it is known to set the confocal parameter of the (non-ideal) pumping beam approximately equal to the absorption length of the crystal, which is called mode-matching. This condition means that the pump beam waist diameter must be above a certain lower limit. The laser-mode waist size is then matched to the pumping-beam waist size. The latter must carefully be optimized; if it is too small, higher-order spatial modes are excited in the laser resonator, and if it is too large, the small-signal gain decreases, the laser threshold increases, and the laser is either not very efficient or does not even reach threshold. Note that mode-matching reduces the saturation fluence in the laser crystal, i.e., improves the laser fluence ratio. (See D. Kopf et al., xe2x80x9cHigh-average-power diode-pumped femtosecond Cr:LiSAF lasersxe2x80x9d, Applied Physics B, vol. 65, pp. 235-243, 1997.)
Nd:vanadate has another key advantage to mode-matchingxe2x80x94it has a very strong and broad pump absorption at the pump wavelength relative to Nd:YAG or Nd:YLF. In addition it can be doped by Nd to levels exceeding 3%, which further allows for increased pump absorption. A typical absorption length in 3%-doped Nd:vanadate is approximately 100 xcexcm. This allows us to achieve mode-matching with a strongly focused pump laser, allowing for a minimum possible pump diameter and therefore a minimum possible laser mode diameter in the laser crystal. This in turn allows for a substantially higher laser fluence in the laser crystal. Combined with the larger cross-section and lower saturation fluence, these effects result in a substantially increased fluence ratio in the laser, and therefore an improved figure of merit against the onset of Q-switching instabilities compared to Nd:YAG or other conventional laser materials.
In addition it is desirable to use a laser diode with the highest possible spatial brightness, where brightness is understood to be the amount of light emitted in proportion to the product of solid-angle of the light times the emitting aperture area. For a given wavelength and power level, the highest brightness light is xe2x80x9cdiffraction limitedxe2x80x9d, which corresponds to light with the minimum possible solid angle from a given emitting area. This is also characterized by the M-squared factor (M2) (see for example M. W. Sasnett, xe2x80x9cPropagation of multimode laser beamsxe2x80x94the M2 factor,xe2x80x9d in The Physics and technology of laser resonators, D. R. Hall, P. E. Jackson, Eds., NY 1989, pp. 132-142) When M2=1, the light is diffraction limited. Larger values of M2 indicate how many times the light is above the diffraction limit. A representative example of current state-of-the-art for high brightness laser diodes is a device emitting at 808 nm, giving up to 2 W of average power from an aperture 100 xcexcm (in the sagittal plane) by 1 xcexcm (in the tangential plane), with a beam divergence of about 10xc2x0 by 35xc2x0 respectively, resulting in an M2 factor of about 20 by 1, respectively. High brightness laser diodes allow the conditions in Eq. (2) to be more easily achieved and/or higher output coupling of the laser resulting in higher average output power.
These results can be generalized. It is possible to achieve higher laser repetition rates with a passively mode-locked laser system by decreasing the laser mode size as much as possible in the laser crystal as limited by mode-matching of the pump diode, choosing the highest brightness pump diode available, decreasing the laser saturation fluence by choosing a laser material with the maximum possible cross-section and a large pump absorption coefficient, then maximizing the fluence on the absorber, and minimizing the saturation fluence of the absorber if possible.
The remaining possibility is to decrease the modulation depth of the absorber xcex94R. As mentioned, the key limitation here is the minimum modulation depth required to start and sustain mode locking. Unfortunately there is not yet a simple analytical form to evaluate the minimum modulation depth required for starting the mode locking process. However it has been observed that the starting threshold depends on several effects, such as the laser design and how many times it is pumped above threshold, and the effect of spatial hole burning due to the positioning of the laser material in the laser cavity (B. Braun et al., xe2x80x9cContinuous-wave mode-locked solid-state lasers with enhanced spatial hole-burning, Part I: Experiments,xe2x80x9dApplied Physics B, vol. 61, pp. 429-437, 1995). We will discuss each of these effects in the following paragraphs.
In general we can state that passive mode locking is more robust (i.e., operates with a faster build-up time and less effects from outside perturbations) when the laser is pumped as hard as possible, i.e., as many times over threshold as can be achieved with the available pump power. This can be understood in the sense that the laser can more quickly respond to changes in its intracavity intensity when pumped more times above threshold.
The effect of spatial hole burning also plays an important role in decreasing the self-starting threshold in passive mode locking. This effect is well described in B. Braun et al., xe2x80x9cContinuous-wave mode-locked solid-state lasers with enhanced spatial hole-burning, Part I: Experiments,xe2x80x9d Applied Physics B, vol. 61, pp. 429-437, 1995. We provide a brief summary of the key points here.
First consider a laser cavity which for the moment is running in continuous-wave mode (i.e., the mode-locking element has been removed, but otherwise it is a typical laser cavity for mode-locked operation). Basically, for laser systems where the gain element is placed substantially at one end of the laser cavity, the frequency separation of the free-running laser modes due to the spatial hole-burning is substantially increased compared to laser systems where the gain is substantially away (at least a few centimeters typically) from the cavity end. For example, in a typical xe2x80x9cgain-in-the-middlexe2x80x9d laser system, the spacing of the free-running modes is only one or a few times the longitudinal mode-spacing (free-spectral range FSR of the cavity=c/2L), e.g., on the order of a few hundred megahertz in a typical mode-locked laser with a FSR of 100 MHz. In a typical xe2x80x9cgain-at-the-endxe2x80x9d laser system, the spacing of the free-running laser modes is many times the FSR of the cavity, typically 100 to 200 times the FSR, e.g., approximately 20 GHz in a typical xe2x80x9cgain-at-the-endxe2x80x9d mode locked laser with a FSR of 100 MHz.
These xe2x80x9cfree-running longitudinal modesxe2x80x9d can assist in the starting and build-up process in passive mode-locking. One can describe the process as follows. In an ideal, homogeneously-broadened laser (approximately similar to the xe2x80x9cgain-in-the-middlexe2x80x9d laser), the laser begins operating with a single (or a few closely-spaced) mode(s). After this mode strikes the saturable absorber, it is modulated such that other longitudinal modes are xe2x80x9cseededxe2x80x9d and begin to grow. Each of these modes is in turn modulated and seeds other adjacent longitudinal modes. This process continues to grow, seeding more and more longitudinal modes, until it reaches a steady state when there is a balance reached between the modulation depth of the absorber and the fall-off in the gain of the laser transition for the modes farthest separated in frequency from the center of the laser transition. This steady-state also sets the final operating bandwidth of the mode-locked laser and thus the minimum operating pulse width.
However, in a xe2x80x9cgain-at-the-endxe2x80x9d laser, this mode-locking process is enhanced due to the initial widely-spaced free-running modes. Instead of starting with just one mode running, each of the free-running modes is modulated by the absorber, seeding adjacent laser modes, until these grow to overlap the other modes which started from a neighboring group. Because the mode locking process does not need to fill as much frequency space compared to the ideal homogeneously broadened case, the mode-locking build-up time is decreased, and the steady-state bandwidth of the system is increased, which results in shorter pulsewidths. This has been experimentally verified in the above Braun reference.
For high-repetition rate lasers, it is advantageous to use the xe2x80x9cgain-at-the-endxe2x80x9d effect to reduce the self-starting threshold, allowing the modulation depth of the absorber to be further decreased, and improving the threshold for the Q-switching according to Eq. (2).
It is also possible to design the frequency-spacing of the free-running modes in a xe2x80x9cgain-at-the-endxe2x80x9d cavity using the length and doping of the crystals, i.e. xcex94f=c/2nlg, where xcex94f is the frequency separation of the modes, c is the speed of light, n is the index of refraction of the crystal, and lg is the length of the crystal where the laser mode is located. It is then possible to design the mode-spacing to be a multiple of the FSR (i.e., repetition rate) of the laser cavity. This should further enhance the self-starting and passive mode locking due to a stronger overlap of the neighboring longitudinal modes. For example, it would be possible to design a 5 GHz repetition rate laser with a crystal length and doping chosen to give a free-running mode-spacing of substantially 20 GHz. This would mean that the fourth mode from a given free-running mode would substantially overlap another free-running mode, enhancing the mode-locking process. If this was chosen poorly, for example a free-running mode-spacing of 22.5 GHz, then the modes would not overlap until the eighth mode away from a given free-running mode. This technique may be especially useful as the laser repetition rate increases above the range of 10 GHz, where the self-starting and Q-switching criteria become more difficult, and where the repetition rate is approaching (within approximately a factor of two) of the free-running mode separation.
We typically use modulation depths in the range of 0.5% to 1% to achieve self-starting while still avoiding the Q-switching instability limit for lasers in the sub-gigahertz range. Optimization of the laser crystal for spatial hole burning to use the above effect will allow for reduced modulation depths from the SESAM and may allow for reliable self-starting with modulation depths substantially below 0.5%, which would be advantageous for repetition rates well above 1 GHz.